3!6!
5!7!
Evaluate and simplify

Answer:

Step-by-step explanation:

The ladder will reach a Height of approximately 43.46 meters on the building.

To find out how high on the building the ladder will reach, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, the ladder forms a right triangle with the ground and the building. The base of the ladder is 7 meters, the height of the building is what we need to find, and the length of the ladder is given as 44 meters.

Using the Pythagorean theorem, we can set up the equation:

(Height of the building)^2 + 7^2 = 44^2

Simplifying the equation, we have:

(Height of the building)^2 + 49 = 1936

Subtracting 49 from both sides, we get:

(Height of the building)^2 = 1887

To find the height of the building, we take the square root of both sides:

Height of the building = √1887

Calculating the square root of 1887, we find that the height of the building is approximately 43.46 meters.

Therefore, the ladder will reach a height of approximately 43.46 meters on the building.

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The product of two numbers is minus 28/27 if one number is (-4/9), then the other number is -7/3.

let x=(-4/9) be the first no. and y be the second no.

then, according to the question,

x × y=-28/27

-4/9 × y=-28/27

y=-28/27 × -9/4

y= -7/3

then second no. i.e y=-7/3

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Answer:

third option

Step-by-step explanation:

under a counterclockwise rotation of 90° about the origin

a point (x, y ) → (y, - x )

Then

A (- 1, - 2 ) → (- 2, - (- 1) ) → (- 2, 1 )

B (2, - 2 ) → (- 2, - 2 )

C (1, - 4 ) → (- 4, - 1 )

The new coordinates are (d) A = (2, -1) B = (2, 2) and C = (4, 1)

From the question, we have the following parameters that can be used in our computation:

The figure,

Where, we have

A = (-1, -2)

B = (2, -2)

C = (1, -4)

The rule of 90 degrees counterclockwise is

(x, y) = (-y, x)

Using the above as a guide, we have the following:

A = (2, -1)

B = (2, 2)

C = (4, 1)

Hence, the new coordinates are (d) A = (2, -1) B = (2, 2) and C = (4, 1)

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The answer is:

x =8

Work/explanation:

For now, I will focus on the left side, and use the distibutive property:

[tex]\sf{4(x+3)+2x=60}[/tex]

[tex]\sf{4x+12+2x=60}[/tex]

[tex]\sf{6x+12=60}[/tex]

Subtract 12 on each side

[tex]\sf{6x=48}[/tex]

Divide:

[tex]\sf{x=8}[/tex]

Answer:

b = 4

Step-by-step explanation:

4/6 is equivalent to 2/3, so with the slopes and y-intercepts being the same, there will be infinitely many solutions

Labeling variables and expressions clearly is important because it helps ensure clarity, precision, and understanding in mathematical or scientific contexts.

Here are a few reasons why it is important to label variables and expressions:

Avoid confusion: Clear labeling prevents confusion, especially when dealing with complex equations or multiple variables. It helps distinguish between different quantities and ensures that each variable represents a specific aspect of the problem or situation.

Enhance communication: Precise labeling allows for effective communication of mathematical ideas and concepts. It helps convey information accurately to others, whether it's through written work, presentations, or discussions.

Promote understanding: By labeling variables and expressions clearly, it becomes easier for both the reader and the writer to understand the meaning and purpose behind each term. It aids in interpreting the mathematical relationships and allows for a better grasp of the overall problem or equation.

Enable error detection: When variables and expressions are properly labeled, it becomes easier to identify any errors or inconsistencies in calculations or equations. It allows for a systematic review and helps catch mistakes in formulas, units, or overall logic.

Facilitate problem-solving: Clear labeling enables effective problem-solving by providing a clear framework for approaching mathematical or scientific challenges. It helps organize information, track relevant variables, and establish logical connections between different components of a problem.

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Answer:

To find the expression for (s+t)(x), we simply add the two functions s(x) and t(x): (s + t)(x) = s(x) + t(x) = 4x - 5 + x + 6 = 5x + 1

To find the expression for (s•t)(x), we multiply the two functions s(x) and t(x): (s•t)(x) = s(x) * t(x) = (4x - 5) * (x + 6) = 4x^2 + 19x - 30

To evaluate (s-t)(2), we substitute 2 for x in the expression for (s-t)(x): (s-t)(2) = s(2) - t(2) = (4*2 - 5) - (2 + 6) = 3 - 8 = -5

Therefore, the expression for (s+t)(x) is 5x+1, the expression for (s•t)(x) is 4x^2+19x-30, and (s-t)(2) is -5.

Step-by-step explanation:

The sprinter will complete the race in approximately 17.07 seconds.

To calculate the time for the race, we need to consider two parts: the acceleration phase and the constant velocity phase.

Acceleration Phase:

The acceleration of the sprinter is 1.64 m/s², and the distance covered during this phase is 25 m. We can use the equation of motion to calculate the time taken during acceleration:

v = u + at

Here:

v = final velocity (which is the velocity at the end of the acceleration phase)

u = initial velocity (which is 0 since the sprinter starts from rest)

a = acceleration

t = time

Rearranging the equation, we have:

t = (v - u) / a

Since the sprinter starts from rest, the initial velocity (u) is 0. Therefore:

t = v / a

Plugging in the values, we get:

t = 25 m / 1.64 m/s²

Constant Velocity Phase:

Once the sprinter reaches the end of the acceleration phase, the velocity remains constant. The remaining distance to be covered is 100 m - 25 m = 75 m. We can calculate the time taken during this phase using the formula:

t = d / v

Here:

d = distance

v = velocity

Plugging in the values, we get:

t = 75 m / (v)

Since the velocity remains constant, we can use the final velocity from the acceleration phase.

Now, let's calculate the time for each phase and sum them up to get the total race time:

Acceleration Phase:

t1 = 25 m / 1.64 m/s²

Constant Velocity Phase:

t2 = 75 m / v

Total race time:

Total time = t1 + t2

Let's calculate the values:

t1 = 25 m / 1.64 m/s² = 15.24 s (rounded to two decimal places)

Now, we need to calculate the final velocity (v) at the end of the acceleration phase. We can use the formula:

v = u + at

Here:

u = initial velocity (0 m/s)

a = acceleration (1.64 m/s²)

t = time (25 m)

Plugging in the values, we get:

v = 0 m/s + (1.64 m/s²)(25 m) = 41 m/s

Now, let's calculate the time for the constant velocity phase:

t2 = 75 m / 41 m/s ≈ 1.83 s (rounded to two decimal places)

Finally, let's calculate the total race time:

Total time = t1 + t2 = 15.24 s + 1.83 s ≈ 17.07 s (rounded to two decimal places)

Therefore, the sprinter will complete the race in approximately 17.07 seconds.

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Entry Point 1 focuses on pattern recognition and rule identification, while Entry Point 2 emphasizes pattern continuation and application.

One mathematical task suitable for intermediate phase learners that can have at least two entry points is a problem involving geometric patterns and sequences. Here's an example:

Problem: Consider the following geometric pattern:

□

□ □

□ □ □

□ □ □ □

Entry Point 1: Identifying the Rule

Ask students to examine the pattern and determine the rule for the number of squares in each row. Encourage them to look for patterns, count the number of squares in each row, and think about how it changes as the rows progress. The entry point here is to observe the pattern and identify the rule that governs the number of squares in each row.

Entry Point 2: Extending the Pattern

Provide students with the first few rows of the pattern and ask them to continue the pattern for a certain number of rows. For example, give them the first four rows and ask them to extend the pattern for three more rows. The entry point here is to extend the pattern by applying the identified rule.

By providing these two entry points, students can engage in different levels of thinking. Entry Point 1 focuses on pattern recognition and rule identification, while Entry Point 2 emphasizes pattern continuation and application.

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The equation of the parabola is:

y = (x + 4)² - 3

We can see that we have the graph of a parabola.

Remember that the vertex form of a parabola whose leading coefficient is a and whose vertex is (h, k) is:

y = a*(x - h)² + k

Here we can see that the vertex is at (-4, -3)

So we can write:

y = a*(x + 4)² - 3

We also can see that the function passes through (-3, -2), replacing that we will get:

-2 = a*(-3 + 4)² - 3

-2 = a - 3

-2 + 3 = a

1 = a

The equation is:

y = (x + 4)² - 3

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Step-by-step explanation:

The ONE  solution to this system of two lines is the point where the two lines cross   at   (1,4)

Answer:

(5, 4)

Step-by-step explanation:

(x + 2, y + 3)

(3  + 2, 1 + 3)

(5, 4)

Answer:

B' (5, 4 )

Step-by-step explanation:

the translation rule (x, y ) → (x + 2, y + 3 )

means add 2 to the original x- coordinate and add 3 to the original y- coordinate , then

B (3, 1 ) → B' (3 + 2, 1 + 3 ) → B' (5, 4 )

To determine the number of minutes you would have to exercise each day to have a resting heart rate of 60 beats per minute, we need to consider the relationship between exercise and heart rate.

Regular exercise can help lower resting heart rate as it strengthens the cardiovascular system. The American Heart Association recommends engaging in moderate-intensity aerobic exercise for at least 150 minutes per week to maintain cardiovascular health.

If we assume that you exercise evenly throughout the week, we can calculate the daily exercise time as follows:

150 minutes per week ÷ 7 days = approximately 21.43 minutes per day.

Therefore, if you exercise for approximately 21.43 minutes per day, it can contribute to maintaining a healthy resting heart rate. It's important to note that individual results may vary, and consulting with a healthcare professional is always recommended before starting or modifying an exercise routine.

However, it's crucial to understand that exercise alone may not be the sole factor affecting resting heart rate. Other factors, such as genetics, overall health, stress levels, and lifestyle choices, can also influence heart rate. Additionally, achieving a resting heart rate of 60 beats per minute may not be feasible or suitable for everyone, as the ideal range can vary based on individual circumstances.

Therefore, while regular exercise can be beneficial for cardiovascular health, it's essential to consider personalized factors and consult with a healthcare professional for tailored advice on achieving and maintaining a healthy resting heart rate.

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Answer:

Step-by-step explanation:

Step 1:First Make 3/4 and 5/6 into like fractions.Find the L.C.M of 4 and  

            6,which is 12.

            3*3/4*3=9/12

             5*2/6*2=10/12

Step 2:Subtract 9/12 from 10/12,which is 1/12.

Answer:

A) 12 inches by 20 inches

Step-by-step explanation:

Dilation of a scale factor means to increase by a factor of 4.
That basically mean multiply the object by 4.

Therefore 3 inches x 4 = 12 inches

And 5 inches x 4 = 20

The function [tex] \sf f(x) [/tex] is defined as:

[tex] \sf f(x) = x^3 + \frac{3}{2}x^2 - 6x + 10 [/tex]

To find the stationary points of [tex] \sf f [/tex], we need to find the values of [tex] \sf x [/tex] where the derivative of [tex] \sf f(x) [/tex] is equal to zero.

First, let's find the derivative of [tex] \sf f(x) [/tex]:

[tex] \sf f'(x) = 3x^2 + 3x - 6 [/tex]

To find the stationary points, we set [tex] \sf f'(x) = 0 [/tex] and solve for [tex] \sf x [/tex]:

[tex] \sf 3x^2 + 3x - 6 = 0 [/tex]

We can factor the quadratic equation as follows:

[tex] \sf 3(x^2 + x - 2) = 0 [/tex]

Now, we solve for [tex] \sf x [/tex] by factoring further:

[tex] \sf 3(x + 2)(x - 1) = 0 [/tex]

This gives us two solutions: [tex] \sf x = -2 [/tex] and [tex] \sf x = 1 [/tex].

So, the stationary points of [tex] \sf f(x) [/tex] are [tex] \sf x = -2 [/tex] and [tex] \sf x = 1 [/tex].

To determine the intervals where [tex] \sf f(x) [/tex] is increasing, we need to analyze the sign of the derivative [tex] \sf f'(x) [/tex] in different intervals. We can use the values of [tex] \sf x = -2 [/tex], [tex] \sf 1 [/tex], and any other value between them.

For [tex] \sf x < -2 [/tex], we choose [tex] \sf x = -3 [/tex] as a test point:

[tex] \sf f'(-3) = 3(-3)^2 + 3(-3) - 6 = 12 > 0 [/tex]

For [tex] \sf -2 < x < 1 [/tex], we choose [tex] \sf x = 0 [/tex] as a test point:

[tex] \sf f'(0) = 3(0)^2 + 3(0) - 6 = -6 < 0 [/tex]

For [tex] \sf x > 1 [/tex], we choose [tex] \sf x = 2 [/tex] as a test point:

[tex] \sf f'(2) = 3(2)^2 + 3(2) - 6 = 18 > 0 [/tex]

From the above analysis, we can conclude that [tex] \sf f(x) [/tex] is increasing in the intervals [tex] \sf (-\infty, -2) [/tex] and [tex] \sf (1, \infty) [/tex].

To find the inflection point of [tex] \sf f [/tex], we need to determine where the concavity changes. This occurs when the second derivative of [tex] \sf f(x) [/tex] changes sign.

The second derivative of [tex] \sf f(x) [/tex] is:

[tex] \sf f''(x) = 6x + 3 [/tex]

To find the inflection point, we set [tex] \sf f''(x) = 0 [/tex] and solve for [tex] \sf x [/tex]:

[tex] \sf 6x + 3 = 0 [/tex]

[tex] \sf 6x = -3 [/tex]

[tex] \sf x = -\frac{1}{2} [/tex]

Therefore, the inflection point of [tex] \sf f(x) [/tex] is [tex] \sf x = -\frac{1}{2} [/tex].

Answer:

C)  $1.93

Step-by-step explanation:

To calculate the additional amount paid for using the credit card instead of cash, we need to consider the interest charges incurred over one month.

The annual interest rate is 16.99%.

Calculate the monthly interest rate by dividing the annual interest rate by 12 (number of months in a year):

[tex]\textsf{Monthly interest rate} = \dfrac{16.99\%}{12} =1.4158333...\%=0.014158333...[/tex]

Now we can calculate the interest charged on the annual subscription fee for one month:

[tex]\begin{aligned}\textsf{Interest charged}& = \textsf{Annual subscription fee} \times \textsf{Monthly interest rate}\\\\&=\$135.99 \times 0.014158333...\\\\&=1.92539175\\\\&= \$1.93\; \textsf{(nearest cent)}\end{aligned}[/tex]

Therefore, the additional amount paid instead of using cash is approximately $1.93.

Answer:

$1.93

Step-by-step explanation:

The first step is to find out how much interest will accrue in one month on the annual fee of $135.99.

[tex]\rm\implies{Interest = \dfrac{APR \times Balance}{12}}[/tex]

Substitute the given values into the formula:

[tex]\begin{aligned}\rm\implies Interest& =\rm \dfrac{16.99 \times 135.99}{12}\\&=\rm\dfrac{23.104701}{12}\\& \approx \boxed{\rm{\$1.93}}\end{aligned}[/tex]

[tex]\therefore[/tex] The subscriber will have paid an additional $1.93 in interest charges by using their credit card instead of paying in cash.

Answer:

Step-by-step explanation:

Perimeter:  

     [tex]P=(x+y+z) \ cm[/tex]

Semi-perimeter:

     [tex]SP=\frac{1}{2} (x+y+z) \ cm[/tex]

Answer:

See below

Step-by-step explanation:

Part A

[tex]f(g(x))=f(\frac{x}{3})=3(\frac{x}{3})=x\\g(f(x))=g(3x)=\frac{3x}{3}=x[/tex]

Since BOTH [tex]f(g(x))=x[/tex] and [tex]g(f(x))=x[/tex], then [tex]f[/tex] and [tex]g[/tex] are inverses of each other

Part B

[tex]f(g(x))=f(\frac{x+1}{2})=2(\frac{x+1}{2})+1=x+1+1=x+2\\g(f(x))=g(2x+1)=\frac{(2x+1)+1}{2}=\frac{2x+2}{2}=x+1[/tex]

Since BOTH [tex]f(g(x))\neq x[/tex] and [tex]g(f(x))\neq x[/tex], then [tex]f[/tex] and [tex]g[/tex] are NOT inverses of each other

The results of the one-way ANOVA suggest that the fast-pass method has a significant impact on the average number of people in queue.

The F-statistic for this example is 3.53. The p-value for the one-way ANOVA is calculated using the F-distribution. The p-value for this example is 0.029.

The p-value is less than the significance level of 0.05, so we can reject the null hypothesis. This means that there is sufficient evidence to conclude that the average number of people in queue is not the same for all three fast-pass methods.

The results of the one-way ANOVA suggest that the fast-pass method has a significant impact on the average number of people in queue. Specifically, Method C has the lowest average number of people in queue, followed by Method A and then Method B.

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So the probability of getting one head and one tail is 50%.since there are no other possible outcomes.

When we flip a coin, there are two possible outcomes: heads (H) or tails (T). Tossing a coin is a random experiment that follows a binomial distribution since each toss is independent, and there are only two possible outcomes.

We can use the binomial probability formula to find the probability of a specific event happening.Here are the answers to each part of the question:b) TT (two tails)

To calculate the probability of getting two tails in a row, we multiply the probability of getting tails on the first toss by the probability of getting tails on the second toss.

P(TT) = P(T) x P(T) = 0.5 x 0.5 = 0.25 = 25%So the probability of getting two tails in a row is 25%.c) HT and TH (one head and one tail)To calculate the probability of getting one head and one tail,

we multiply the probability of getting a head on the first toss by the probability of getting a tail on the second toss, and then add it to the probability of getting a tail on the first toss and a head on the second toss.

P(HT or TH) = P(H) x P(T) + P(T) x P(H) = 0.5 x 0.5 + 0.5 x 0.5 = 0.5 = 50%

Note that the probability of getting two heads is the same as the probability of getting two tails (25%), and the probability of getting any combination of heads and tails is 100% (1),

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To find the percentage of racers who did not complete the marathon, we need to calculate the proportion of racers who gave up or were disqualified out of the total number of racers who started the marathon.

The number of racers who did not complete the marathon is the sum of those who gave up and those who were disqualified:

Number of racers who did not complete = Number who gave up + Number who were disqualified

                                     = 23 + 3

                                     = 26

Now, we can calculate the percentage using the formula:

Percentage = (Number who did not complete / Total number who started) * 100

Percentage = (26 / 240) * 100

Percentage ≈ 10.83%

Therefore, approximately 10.83% of the racers did not complete the marathon.

This percentage represents the portion of racers who were unable to finish the race due to various reasons such as fatigue, injury, or disqualification. It highlights the challenges and demands of participating in a marathon and the determination required to complete the race.

The percentage provides a quantitative measure of the proportion of racers who were not able to reach the finish line, giving an understanding of the attrition rate in the marathon.

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The number of triangles in the diagram that can be mapped to one another by similarity transformations include the following: D. 3.

In Mathematics and Geometry, two triangles are said to be similar when the ratio of their corresponding side lengths are equal and their corresponding angles are congruent.

Additionally, the lengths of three (3) pairs of corresponding sides or corresponding side lengths are proportional to the lengths of corresponding altitudes when two (2) triangles are similar.

Based on the side, side, side (SSS) similarity theorem, we can logically deduce the following congruent and similar triangles:

ΔABC

ΔDEF

ΔGHI

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Answer:

Step-by-step explanation:

its is 14

Adult admission costs $18, and child admission costs $5.To check:2(5) + 18 = 28, which is true3(5) + 18 = 48, which is also true.

Let's begin by letting x be the cost of adult admission, and y be the cost of a child admission, so we have two equations:2y + x = 28
3y + x = 48

Subtracting the first equation from the second one, we get:3y + x - (2y + x) = 48 - 28y = 20Therefore, we can solve for x by plugging in y=5 in either equation.

Using the first equation, we have:2(5) + x = 28x = 18

Thus, adult admission costs $18, and child admission costs $5.To check:2(5) + 18 = 28, which is true3(5) + 18 = 48, which is also true.

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Answer:

g(- 5) = 3

Step-by-step explanation:

to find g(- 5) substitute x = - 5 into g(x)

g(- 5) = - (- 5) - 2 = 5 - 2 = 3

For an input of 2, the output value of the function include the following: D. after 2 hours, the cost is $6.

In Mathematics and Geometry, a linear function refers to a type of function whose equation is graphically represented by a straight line on the xy-plane or cartesian coordinate.

This ultimately implies that, a linear function has the same (constant) slope and it is typically used for uniquely mapping an input variable to an output variable, which both increases simultaneously.

In this context, we can logically deduce that the total cost of parking after 2 hours is equal to 6 dollars based on the graph shown above.

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The annual rates of return for Asman and Salinas are as follows: Asman - 22.22%, 9.09%, 16.67%; Salinas - 60.00%%.

1. To calculate the annual rates of return, we need to determine the percentage change in stock prices from one time period to another.

2. For Asman stock, the price data is as follows:

  - Time 1: $30

  - Time 2: $11

  - Time 3: $29

  - Time 4: $12

3. The rate of return you would have earned on Asman stock from time 1 to time 2 can be calculated using the formula:

  [(Ending Price - Beginning Price) / Beginning Price] * 100

  Substituting the values, we get:

  [(11 - 30) / 30] * 100 = -63.33% (rounded to two decimal places)

4. The rate of return you would have earned on Asman stock from time 2 to time 3 can be calculated similarly:

  [(29 - 11) / 11] * 100 = 163.64% (rounded to two decimal places)

5. The rate of return you would have earned on Asman stock from time 3 to time 4 can be calculated:

  [(12 - 29) / 29] * 100 = -58.62% (rounded to two decimal places)

6. For Salinas stock, the price data is as follows:

  - Time 1: $30

  - Time 2: $12

  - Time 3: $31

  - Time 4: $34

7. The rate of return you would have earned on Salinas stock from time 1 to time 2 can be calculated:

  [(12 - 30) / 30] * 100 = -60.00% (rounded to two decimal places)

8. Therefore, the annual rates of return for Asman and Salinas are as follows:

  - Asman: -63.33%, 163.64%, -58.62% (rounded to two decimal places)

  - Salinas: -60.00% (rounded to two decimal places)

9. The annual rates of return indicate the percentage change in the value of the stock over a one-year period. A positive rate of return indicates a gain, while a negative rate of return indicates a loss. These figures help investors assess the performance of their investments and make informed decisions.

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The arborist will make approximately $28.65.

First, let's draw an image of the problem:

Using the Pythagorean Theorem, we can find the diagonal measurement (d) from the top of the tree to the end of the shadow:

d² = h² + s²

Substituting the given values:

d² = 15² + 8²

d² = 225 + 64

d² = 289

Taking the square root of both sides:

d = √289

d = 17 feet

Now, let's convert the measurements from feet to meters using the conversion ratio:

1 meter / 3.3 feet

Height in meters: 15 feet [tex]\times[/tex] (1 meter / 3.3 feet) ≈ 4.55 meters

Shadow in meters: 8 feet [tex]\times[/tex] (1 meter / 3.3 feet) ≈ 2.42 meters

Diagonal in meters: 17 feet [tex]\times[/tex] (1 meter / 3.3 feet) ≈ 5.15 meters

To calculate the arborist's earnings, we need to find the square area (in square meters) of the tree and its shadow:

Area = Height [tex]\times[/tex] Shadow

Area = 4.55 meters [tex]\times[/tex] 2.42 meters

Area ≈ 11.02 square meters

Finally, the arborist will earn:

Earnings = Area [tex]\times[/tex] $2.60 per square meter

Earnings = 11.02 square meters [tex]\times[/tex] $2.60 per square meter

Earnings ≈ $28.65  

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